The reason why we studied small values of \(N\)  in this work
is to generate results that can be analytically checked up to a good degree.
Obviously, for larger systems of particles, it is not possible
to find the minimum energy configuration by using analytical tools.
For such a scenario, the entire approach is considerably
more difficult and should be fully numerical.
In order to get glimpses of the behavior of the system at larger values of \(N\) ,
we carried out additional numerical calculations using the
simulated annealing method [1]}
in order to find the minimum energy configuration for some of the largest
values of \(N\)  that we could reliably simulate.
